# Bayes’ Theorem Examples

## Example 1

Historical Data.

P(cavity) = 0.1                    — 10%

P(toothache) = 0.05         — 5%

P(Cavity | toothache) = 0.8   — 80%

Future Prediction:

P(toothache | cavity)  = ?

using bays law à P(B|A) = {P(A|B)P(B)}/P(A)

P(toothache | cavity)  = {P(cavity | toothache) P(toothache) } / P(cavity)

=             (0.8*0.05)/0.1 = 0.4

## Example 2

If HIV has a prevalence of 3% in San Francisco, and a particular HIV test has a false positive rate of .001 and a false negative rate of .01, what is the probability that a random person who tests positive is actually infected (also known as “positive predictive value”)?

### Solution of Example 2

We have to find the probability of hiv+ given the report is positive.

Using Bayes theorem
P(hiv+|t+) = P(t+|hiv+)P(hiv+) / P(t+)

Now,

false positive  P(t+|hiv-) = .001

P(t-|hiv-) =1-.001 =  .999

false negative = P(t-|hiv+) = .01
P(t+|hiv+) = 1-.01 = .99

P(t+) = P(t+&hiv+) + P(t+&hiv-)
=P(t+|hiv+)P(hiv+) + P(t+|hiv-)P(hiv-)
= .99*.03 + .001*.97

Putting all these values in equation
P(hiv+|t+) = P(t+|hiv+)P(hiv+) / P(t+)

= (.99 * .03)/{.99*.03 + .001*.97}

= .968   ===> 96.8%

Bayes’s Theorem Example 1

Bayes’s Theorem Example 2

Bayes’s Theorem Example 3